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Beyond Taylor: Divergence-Based Functional Expansions and Their Application to Numerical Integration

Junping Wang

TL;DR

This work proposes a divergence-based functional expansion as an alternative to classical Taylor expansions for high-order numerical integration on arbitrary flat-faced polytopes. By rewriting the integrand as a sum of divergence terms and repeatedly applying the Divergence Theorem, volume integrals are transformed into boundary integrals and reduced dimensionally to facets and ultimately vertices. The paper delivers (i) a general JW-type expansion in $\mathbb{R}^n$, (ii) a complex-shift enhancement that yields real high-order quadrature rules exact for low-degree polynomials, and (iii) rigorous affine-geometry results for surface transformations, normals, and measures, enabling robust, boundary-based quadrature schemes. Together, these contributions provide a computationally efficient, structure-preserving alternative to tessellation-based methods for integrating over general flat-faced polytopes and their surfaces, with explicit formulas for volumes, centers of mass, and polynomial integrals derived from boundary data.

Abstract

This paper introduces a new functional expansion framework that extends classical ideas beyond the Taylor series. Unlike traditional Taylor expansions based on local polynomial approximations, the proposed approach arises from exact differential identities that link a function and its derivatives through polynomial weight factors. This formulation expresses smooth functions via divergence-based relations connecting derivatives of all orders with systematically scaled polynomial coefficients. This framework provides a natural foundation for constructing high-order numerical quadrature formulas, particularly for multi-dimensional domains. By exploiting the divergence structure, volume integrals are systematically transformed into boundary integrals using the Divergence Theorem, recursively reducing the integration domain from an $n$-dimensional body to its $(n-1)$-dimensional facets, and ultimately to its vertices. The article further enhances the framework's accuracy by introducing a complex-shift technique. It is demonstrated that by positioning the expansion center at specific roots of unity in the complex plane, lower-order error terms are annihilated, yielding high-order real-valued quadrature rules with minimal function evaluations. Additionally, a rigorous geometric analysis of the affine transformations required for surface integration is provided, deriving explicit formulas for the transformation of normal vectors and surface measures. The proposed method offers a robust, systematic, and computationally efficient alternative to tessellation-based quadrature for arbitrary flat-faced polytopes.

Beyond Taylor: Divergence-Based Functional Expansions and Their Application to Numerical Integration

TL;DR

This work proposes a divergence-based functional expansion as an alternative to classical Taylor expansions for high-order numerical integration on arbitrary flat-faced polytopes. By rewriting the integrand as a sum of divergence terms and repeatedly applying the Divergence Theorem, volume integrals are transformed into boundary integrals and reduced dimensionally to facets and ultimately vertices. The paper delivers (i) a general JW-type expansion in , (ii) a complex-shift enhancement that yields real high-order quadrature rules exact for low-degree polynomials, and (iii) rigorous affine-geometry results for surface transformations, normals, and measures, enabling robust, boundary-based quadrature schemes. Together, these contributions provide a computationally efficient, structure-preserving alternative to tessellation-based methods for integrating over general flat-faced polytopes and their surfaces, with explicit formulas for volumes, centers of mass, and polynomial integrals derived from boundary data.

Abstract

This paper introduces a new functional expansion framework that extends classical ideas beyond the Taylor series. Unlike traditional Taylor expansions based on local polynomial approximations, the proposed approach arises from exact differential identities that link a function and its derivatives through polynomial weight factors. This formulation expresses smooth functions via divergence-based relations connecting derivatives of all orders with systematically scaled polynomial coefficients. This framework provides a natural foundation for constructing high-order numerical quadrature formulas, particularly for multi-dimensional domains. By exploiting the divergence structure, volume integrals are systematically transformed into boundary integrals using the Divergence Theorem, recursively reducing the integration domain from an -dimensional body to its -dimensional facets, and ultimately to its vertices. The article further enhances the framework's accuracy by introducing a complex-shift technique. It is demonstrated that by positioning the expansion center at specific roots of unity in the complex plane, lower-order error terms are annihilated, yielding high-order real-valued quadrature rules with minimal function evaluations. Additionally, a rigorous geometric analysis of the affine transformations required for surface integration is provided, deriving explicit formulas for the transformation of normal vectors and surface measures. The proposed method offers a robust, systematic, and computationally efficient alternative to tessellation-based quadrature for arbitrary flat-faced polytopes.
Paper Structure (15 sections, 11 theorems, 128 equations, 1 figure, 1 table)

This paper contains 15 sections, 11 theorems, 128 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The One-Dimensional Case: Functional Expansion
  3. Numerical Integration in One Dimension
  4. Improving Accuracy via a Complex Shift
  5. Four-term Formula via a Complex Shift
  6. Functional Expansion in $n$ Dimensions
  7. Functional Expansion on Flat Surfaces
  8. Numerical Integration in Multidimensional Domains
  9. Volume Calculation
  10. Center of Mass Calculation
  11. Integration of Polynomials
  12. Recursive Trapezoidal Rule in Higher Dimensions
  13. Integration on Flat Surfaces
  14. m-Dimensional Volume of a Parallelepiped
  15. Transformation of Normal Vectors

Key Result

Theorem 2.1

\newlabelthm:finite_expansion Let $m \geqslant 0$ be an integer and $f \in C^{m+1}(\mathbb{R})$. Then Moreover, if $f \in C^\infty(\mathbb{R})$, we have the infinite expansion provided the series converges.

Figures (1)

  • Figure 5.1: Depiction of a flat face $\Sigma \subset \partial T$.

Theorems & Definitions (15)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 5.1
  • Theorem 5.2
  • Theorem 6.1
  • Theorem 7.1
  • ...and 5 more